Fit log returns to F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of
freedom).
xi is the estimated skewness parameter.
For 2011, medium risk data is used in the high risk data set, as no
high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from
2007 to 2023. For 2007 to 2011 (both included) no high risk data is
available.
## vmr vhr pmr phr
## Min. :0.868 Min. :0.849 Min. :0.904 Min. :0.878
## 1st Qu.:1.044 1st Qu.:1.039 1st Qu.:1.042 1st Qu.:1.068
## Median :1.097 Median :1.099 Median :1.084 Median :1.128
## Mean :1.070 Mean :1.085 Mean :1.065 Mean :1.095
## 3rd Qu.:1.136 3rd Qu.:1.160 3rd Qu.:1.107 3rd Qu.:1.182
## Max. :1.168 Max. :1.214 Max. :1.141 Max. :1.208
## mmr mhr
## Min. :0.988 Min. :0.977
## 1st Qu.:1.013 1st Qu.:1.013
## Median :1.085 Median :1.113
## Mean :1.066 Mean :1.087
## 3rd Qu.:1.101 3rd Qu.:1.128
## Max. :1.133 Max. :1.207
## vmrl
## Min. :0.801
## 1st Qu.:1.013
## Median :1.085
## Mean :1.061
## 3rd Qu.:1.128
## Max. :1.193
## vmr vhr pmr phr mmr mhr
## Min. : 0.868 0.849 0.904 0.878 0.988 0.977
## 1st Qu.: 1.044 1.039 1.042 1.068 1.013 1.013
## Median : 1.097 1.099 1.084 1.128 1.085 1.113
## Mean : 1.070 1.085 1.065 1.095 1.066 1.087
## 3rd Qu.: 1.136 1.160 1.107 1.182 1.101 1.128
## Max. : 1.168 1.214 1.141 1.208 1.133 1.207
| Min. : | ranking | 1st Qu.: | ranking | Median : | ranking | Mean : | ranking | 3rd Qu.: | ranking | Max. : | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.988 | mmr | 1.068 | phr | 1.128 | phr | 1.095 | phr | 1.182 | phr | 1.214 | vhr |
| 0.977 | mhr | 1.044 | vmr | 1.113 | mhr | 1.087 | mhr | 1.160 | vhr | 1.208 | phr |
| 0.904 | pmr | 1.042 | pmr | 1.099 | vhr | 1.085 | vhr | 1.136 | vmr | 1.207 | mhr |
| 0.878 | phr | 1.039 | vhr | 1.097 | vmr | 1.070 | vmr | 1.128 | mhr | 1.168 | vmr |
| 0.868 | vmr | 1.013 | mmr | 1.085 | mmr | 1.066 | mmr | 1.107 | pmr | 1.141 | pmr |
| 0.849 | vhr | 1.013 | mhr | 1.084 | pmr | 1.065 | pmr | 1.101 | mmr | 1.133 | mmr |
## cov(vmr, pmr) = -0.001094875
## cov(vhr, phr) = -0.0001730651
##
## AIC: -27.8497
## BIC: -25.58991
## m: 0.0480931
## s: 0.1198426
## nu (df): 3.303595
## xi: 0.03361192
## R^2: 0.993
##
## An R^2 of 0.993 suggests that the fit is extremely good.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 63.16667 percent
## What is the chance of gaining min 25 %? >= 49.33333 percent
## What is the chance of gaining min 50 %? >= 40.16667 percent
## What is the chance of gaining min 90 %? >= 32.66667 percent
## What is the chance of gaining min 99 %? >= 31.5 percent
The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 278.995 kr.
## SD of portfolio index value after 20 years: 123.583 kr.
## Min total portfolio index value after 20 years: 9.812 kr.
## Max total portfolio index value after 20 years: 929.858 kr.
##
## Share of paths finishing below 100: 4.72 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.1724769 0.3205692
Objective function plots
##
## AIC: -34.35752
## BIC: -31.02467
## m: 0.05171176
## s: 0.1149408
## nu (df): 2.706099
## xi: 0.5049945
## R^2: 0.978
##
## An R^2 of 0.978 suggests that the fit is very good.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 58.66667 percent
## What is the chance of gaining min 25 %? >= 47.5 percent
## What is the chance of gaining min 50 %? >= 40.16667 percent
## What is the chance of gaining min 90 %? >= 34 percent
## What is the chance of gaining min 99 %? >= 33 percent
The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 293.384 kr.
## SD of portfolio index value after 20 years: 118.147 kr.
## Min total portfolio index value after 20 years: 0.785 kr.
## Max total portfolio index value after 20 years: 2029.133 kr.
##
## Share of paths finishing below 100: 3.11 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.1842753 0.3193925
Objective function plots
##
## AIC: -21.42488
## BIC: -19.16508
## m: 0.06471454
## s: 0.1499924
## nu (df): 3.144355
## xi: 0.002367034
## R^2: 0.991
##
## An R^2 of 0.991 suggests that the fit is extremely good.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 64.66667 percent
## What is the chance of gaining min 25 %? >= 47.83333 percent
## What is the chance of gaining min 50 %? >= 36.83333 percent
## What is the chance of gaining min 90 %? >= 28 percent
## What is the chance of gaining min 99 %? >= 26.5 percent
The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
##
## Mean portfolio index value after 20 years: 404.855 kr.
## SD of portfolio index value after 20 years: 217.274 kr.
## Min total portfolio index value after 20 years: 0.01 kr.
## Max total portfolio index value after 20 years: 1713.743 kr.
##
## Share of paths finishing below 100: 4.3 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.5302163 0.4155546
Objective function plots
##
## AIC: -33.22998
## BIC: -30.97018
## m: 0.05789224
## s: 0.1234592
## nu (df): 2.265273
## xi: 0.477324
## R^2: 0.991
##
## An R^2 of 0.991 suggests that the fit is extremely good.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 52.83333 percent
## What is the chance of gaining min 25 %? >= 44 percent
## What is the chance of gaining min 50 %? >= 38.83333 percent
## What is the chance of gaining min 90 %? >= 34.66667 percent
## What is the chance of gaining min 99 %? >= 34 percent
The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.
## [1] -0.091256521 -0.003731241 0.027312079 0.045808232 0.059068633
## [6] 0.069575113 0.078454727 0.086316936 0.093536451 0.100370932
## [11] 0.107018607 0.114081432 0.127604387
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.
## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
##
## Mean portfolio index value after 20 years: 321.882 kr.
## SD of portfolio index value after 20 years: 106.531 kr.
## Min total portfolio index value after 20 years: 0.01 kr.
## Max total portfolio index value after 20 years: 1121.574 kr.
##
## Share of paths finishing below 100: 1.94 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.2338345 0.2992717
Objective function plots
##
## AIC: -23.72565
## BIC: -21.46585
## m: 0.08386034
## s: 0.1210107
## nu (df): 3.184569
## xi: 0.01790306
## R^2: 0.964
##
## An R^2 of 0.964 suggests that the fit is very good.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 56.83333 percent
## What is the chance of gaining min 25 %? >= 43.16667 percent
## What is the chance of gaining min 50 %? >= 34.16667 percent
## What is the chance of gaining min 90 %? >= 26.83333 percent
## What is the chance of gaining min 99 %? >= 25.66667 percent
The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 555.222 kr.
## SD of portfolio index value after 20 years: 244.751 kr.
## Min total portfolio index value after 20 years: 0.765 kr.
## Max total portfolio index value after 20 years: 1828.26 kr.
##
## Share of paths finishing below 100: 0.95 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.7617723 0.4255421
Objective function plots
##
## AIC: -36.9603
## BIC: -34.7005
## m: 0.05902873
## s: 0.08757749
## nu (df): 2.772621
## xi: 0.02904471
## R^2: 0.89
##
## An R^2 of 0.89 suggests that the fit is not completely random.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 53.16667 percent
## What is the chance of gaining min 25 %? >= 44.16667 percent
## What is the chance of gaining min 50 %? >= 38.66667 percent
## What is the chance of gaining min 90 %? >= 34.16667 percent
## What is the chance of gaining min 99 %? >= 33.5 percent
The fit suggests big losses for the lowest percentiles, which are not
present in the data.
So the fit is actually a very cautious estimate.
Let’s plot the fit and the observed returns together.
Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 324.055 kr.
## SD of portfolio index value after 20 years: 98.254 kr.
## Min total portfolio index value after 20 years: 3.973 kr.
## Max total portfolio index value after 20 years: 723.752 kr.
##
## Share of paths finishing below 100: 1.13 percent
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 300.815 kr.
## SD of portfolio index value after 20 years: 82.422 kr.
## Min total portfolio index value after 20 years: 42.661 kr.
## Max total portfolio index value after 20 years: 1008.214 kr.
##
## Share of paths finishing below 100: 0.34 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.1398383 0.2595942
Objective function plots
##
## AIC: -24.26084
## BIC: -22.00104
## m: 0.0822419
## s: 0.07129843
## nu (df): 89.86289
## xi: 0.7697502
## R^2: 0.961
##
## An R^2 of 0.961 suggests that the fit is very good.
##
## What is the risk of losing max 10 %? =< 0 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 52.5 percent
## What is the chance of gaining min 25 %? >= 45 percent
## What is the chance of gaining min 50 %? >= 38.33333 percent
## What is the chance of gaining min 90 %? >= 31.16667 percent
## What is the chance of gaining min 99 %? >= 29.83333 percent
The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that the high risk mix provides a much better upside and smaller downside.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 498.653 kr.
## SD of portfolio index value after 20 years: 155.916 kr.
## Min total portfolio index value after 20 years: 150.86 kr.
## Max total portfolio index value after 20 years: 1463.716 kr.
##
## Share of paths finishing below 100: 0 percent
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 479.218 kr.
## SD of portfolio index value after 20 years: 164.197 kr.
## Min total portfolio index value after 20 years: 55.397 kr.
## Max total portfolio index value after 20 years: 1335.395 kr.
##
## Share of paths finishing below 100: 0.12 percent
1e6 paths:
# Down-and-out simulation:
# Probability of down-and-out: 0 percent
#
# Mean portfolio index value after 20 years: 478.339 kr.
# SD of portfolio index value after 20 years: 163.093 kr.
# Min total portfolio index value after 20 years: 2.233 kr.
# Max total portfolio index value after 20 years: 1561.965 kr.
#
# Share of paths finishing below 100: 0.1181 percent
Max vs sum plots for the first four moments:
Parameters
## [1] 1.5927304 0.3361558
Objective function plots
Risk of max loss of x percent for a single period (year).
x values are row names.
| Velliv_m | Velliv_m_l | Velliv_h | PFA_m | PFA_h | mix_m | mix_h | |
|---|---|---|---|---|---|---|---|
| 0 | 21.167 | 17.833 | 19.667 | 11.833 | 14.000 | 12.333 | 12.667 |
| 5 | 12.167 | 9.333 | 12.500 | 5.667 | 8.333 | 5.833 | 3.833 |
| 10 | 7.000 | 5.000 | 8.000 | 3.000 | 5.000 | 2.833 | 0.500 |
| 25 | 1.333 | 0.833 | 2.167 | 0.500 | 1.000 | 0.333 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.167 | Velliv_m | 12.500 | Velliv_h | 8.000 | Velliv_h | 2.167 | Velliv_h | 0 | Velliv_m | 0 | Velliv_m | 0 | Velliv_m |
| 19.667 | Velliv_h | 12.167 | Velliv_m | 7.000 | Velliv_m | 1.333 | Velliv_m | 0 | Velliv_m_l | 0 | Velliv_m_l | 0 | Velliv_m_l |
| 17.833 | Velliv_m_l | 9.333 | Velliv_m_l | 5.000 | Velliv_m_l | 1.000 | PFA_h | 0 | Velliv_h | 0 | Velliv_h | 0 | Velliv_h |
| 14.000 | PFA_h | 8.333 | PFA_h | 5.000 | PFA_h | 0.833 | Velliv_m_l | 0 | PFA_m | 0 | PFA_m | 0 | PFA_m |
| 12.667 | mix_h | 5.833 | mix_m | 3.000 | PFA_m | 0.500 | PFA_m | 0 | PFA_h | 0 | PFA_h | 0 | PFA_h |
| 12.333 | mix_m | 5.667 | PFA_m | 2.833 | mix_m | 0.333 | mix_m | 0 | mix_m | 0 | mix_m | 0 | mix_m |
| 11.833 | PFA_m | 3.833 | mix_h | 0.500 | mix_h | 0.000 | mix_h | 0 | mix_h | 0 | mix_h | 0 | mix_h |
Chance of min gains of x percent for a single period (year).
x values are row names.
| Velliv_m | Velliv_m_l | Velliv_h | PFA_m | PFA_h | mix_m | mix_h | |
|---|---|---|---|---|---|---|---|
| 0 | 78.833 | 82.167 | 80.333 | 88.167 | 86.000 | 87.667 | 87.333 |
| 5 | 63.833 | 65.000 | 69.333 | 71.667 | 76.000 | 71.667 | 70.167 |
| 10 | 40.833 | 36.000 | 53.333 | 32.500 | 59.667 | 35.500 | 46.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.833 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 88.167 | PFA_m | 76.000 | PFA_h | 59.667 | PFA_h | 0.833 | mix_h | 0 | Velliv_m | 0 | Velliv_m |
| 87.667 | mix_m | 71.667 | PFA_m | 53.333 | Velliv_h | 0.000 | Velliv_m | 0 | Velliv_m_l | 0 | Velliv_m_l |
| 87.333 | mix_h | 71.667 | mix_m | 46.000 | mix_h | 0.000 | Velliv_m_l | 0 | Velliv_h | 0 | Velliv_h |
| 86.000 | PFA_h | 70.167 | mix_h | 40.833 | Velliv_m | 0.000 | Velliv_h | 0 | PFA_m | 0 | PFA_m |
| 82.167 | Velliv_m_l | 69.333 | Velliv_h | 36.000 | Velliv_m_l | 0.000 | PFA_m | 0 | PFA_h | 0 | PFA_h |
| 80.333 | Velliv_h | 65.000 | Velliv_m_l | 35.500 | mix_m | 0.000 | PFA_h | 0 | mix_m | 0 | mix_m |
| 78.833 | Velliv_m | 63.833 | Velliv_m | 32.500 | PFA_m | 0.000 | mix_m | 0 | mix_h | 0 | mix_h |
Risk of loss from first to last period.
_a is simulation from estimated distribution of returns
of mix.
_b is mix of simulations from estimated distribution of
returns from individual funds.
_m is medium.
_h is high.
| Velliv_m | Velliv_m_l | Velliv_h | PFA_m | PFA_h | mix_m_a | mix_h_a | mix_m_b | mix_h_b | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 4.72 | 3.11 | 4.30 | 1.94 | 0.95 | 1.13 | 0 | 0.34 | 0.12 |
| 5 | 4.14 | 2.78 | 3.82 | 1.81 | 0.81 | 1.02 | 0 | 0.23 | 0.09 |
| 10 | 3.72 | 2.27 | 3.35 | 1.70 | 0.70 | 0.90 | 0 | 0.13 | 0.09 |
| 25 | 2.42 | 1.52 | 2.44 | 1.23 | 0.47 | 0.55 | 0 | 0.07 | 0.03 |
| 50 | 0.86 | 0.69 | 1.11 | 0.59 | 0.25 | 0.27 | 0 | 0.01 | 0.00 |
| 90 | 0.01 | 0.11 | 0.10 | 0.18 | 0.04 | 0.02 | 0 | 0.00 | 0.00 |
| 99 | 0.00 | 0.01 | 0.01 | 0.06 | 0.01 | 0.00 | 0 | 0.00 | 0.00 |
1e6 simulation paths of mhr_b:
| 0 | 5 | 10 | 25 | 50 | 90 | 99 | |
|---|---|---|---|---|---|---|---|
| prob_pct | 0.118 | 0.095 | 0.076 | 0.036 | 0.008 | 0 | 0 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.72 | Velliv_m | 4.14 | Velliv_m | 3.72 | Velliv_m | 2.44 | Velliv_h | 1.11 | Velliv_h | 0.18 | PFA_m | 0.06 | PFA_m |
| 4.30 | Velliv_h | 3.82 | Velliv_h | 3.35 | Velliv_h | 2.42 | Velliv_m | 0.86 | Velliv_m | 0.11 | Velliv_m_l | 0.01 | Velliv_m_l |
| 3.11 | Velliv_m_l | 2.78 | Velliv_m_l | 2.27 | Velliv_m_l | 1.52 | Velliv_m_l | 0.69 | Velliv_m_l | 0.10 | Velliv_h | 0.01 | Velliv_h |
| 1.94 | PFA_m | 1.81 | PFA_m | 1.70 | PFA_m | 1.23 | PFA_m | 0.59 | PFA_m | 0.04 | PFA_h | 0.01 | PFA_h |
| 1.13 | mix_m_a | 1.02 | mix_m_a | 0.90 | mix_m_a | 0.55 | mix_m_a | 0.27 | mix_m_a | 0.02 | mix_m_a | 0.00 | Velliv_m |
| 0.95 | PFA_h | 0.81 | PFA_h | 0.70 | PFA_h | 0.47 | PFA_h | 0.25 | PFA_h | 0.01 | Velliv_m | 0.00 | mix_m_a |
| 0.34 | mix_m_b | 0.23 | mix_m_b | 0.13 | mix_m_b | 0.07 | mix_m_b | 0.01 | mix_m_b | 0.00 | mix_h_a | 0.00 | mix_h_a |
| 0.12 | mix_h_b | 0.09 | mix_h_b | 0.09 | mix_h_b | 0.03 | mix_h_b | 0.00 | mix_h_a | 0.00 | mix_m_b | 0.00 | mix_m_b |
| 0.00 | mix_h_a | 0.00 | mix_h_a | 0.00 | mix_h_a | 0.00 | mix_h_a | 0.00 | mix_h_b | 0.00 | mix_h_b | 0.00 | mix_h_b |
Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of
mix.
_b is mix of simulations from estimated distribution of
returns from individual funds.
| Velliv_m | Velliv_m_l | Velliv_h | PFA_m | PFA_h | mix_m_a | mix_h_a | mix_m_b | mix_h_b | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 95.28 | 96.89 | 95.70 | 98.06 | 99.05 | 98.87 | 100.00 | 99.66 | 99.88 |
| 5 | 94.67 | 96.40 | 95.17 | 97.80 | 98.93 | 98.72 | 100.00 | 99.55 | 99.83 |
| 10 | 94.00 | 95.89 | 94.73 | 97.56 | 98.78 | 98.54 | 100.00 | 99.39 | 99.82 |
| 25 | 91.57 | 93.96 | 93.21 | 96.68 | 98.42 | 97.85 | 100.00 | 99.01 | 99.72 |
| 50 | 85.87 | 89.95 | 90.33 | 94.73 | 97.50 | 96.23 | 100.00 | 97.42 | 99.31 |
| 100 | 71.62 | 78.11 | 83.11 | 87.55 | 94.60 | 89.72 | 99.66 | 89.73 | 97.46 |
| 200 | 39.14 | 44.61 | 64.53 | 58.58 | 85.28 | 59.42 | 93.18 | 48.34 | 87.18 |
| 300 | 16.10 | 18.01 | 44.83 | 22.36 | 71.62 | 22.28 | 71.24 | 11.58 | 65.20 |
| 400 | 5.20 | 4.56 | 29.08 | 4.45 | 54.84 | 3.77 | 43.90 | 1.27 | 42.17 |
| 500 | 1.45 | 1.16 | 17.33 | 0.59 | 39.30 | 0.22 | 22.71 | 0.07 | 22.12 |
| 1000 | 0.00 | 0.03 | 0.59 | 0.01 | 2.28 | 0.00 | 0.25 | 0.00 | 0.11 |
1e6 simulation paths of mhr_b:
| 0 | 5 | 10 | 25 | 50 | 100 | 200 | 300 | 400 | 500 | 1000 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| prob | 99.882 | 99.854 | 99.824 | 99.686 | 99.301 | 97.513 | 86.912 | 65.992 | 41.486 | 21.693 | 0.086 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 100.00 | mix_h_a | 100.00 | mix_h_a | 100.00 | mix_h_a | 100.00 | mix_h_a | 100.00 | mix_h_a | 99.66 | mix_h_a |
| 99.88 | mix_h_b | 99.83 | mix_h_b | 99.82 | mix_h_b | 99.72 | mix_h_b | 99.31 | mix_h_b | 97.46 | mix_h_b |
| 99.66 | mix_m_b | 99.55 | mix_m_b | 99.39 | mix_m_b | 99.01 | mix_m_b | 97.50 | PFA_h | 94.60 | PFA_h |
| 99.05 | PFA_h | 98.93 | PFA_h | 98.78 | PFA_h | 98.42 | PFA_h | 97.42 | mix_m_b | 89.73 | mix_m_b |
| 98.87 | mix_m_a | 98.72 | mix_m_a | 98.54 | mix_m_a | 97.85 | mix_m_a | 96.23 | mix_m_a | 89.72 | mix_m_a |
| 98.06 | PFA_m | 97.80 | PFA_m | 97.56 | PFA_m | 96.68 | PFA_m | 94.73 | PFA_m | 87.55 | PFA_m |
| 96.89 | Velliv_m_l | 96.40 | Velliv_m_l | 95.89 | Velliv_m_l | 93.96 | Velliv_m_l | 90.33 | Velliv_h | 83.11 | Velliv_h |
| 95.70 | Velliv_h | 95.17 | Velliv_h | 94.73 | Velliv_h | 93.21 | Velliv_h | 89.95 | Velliv_m_l | 78.11 | Velliv_m_l |
| 95.28 | Velliv_m | 94.67 | Velliv_m | 94.00 | Velliv_m | 91.57 | Velliv_m | 85.87 | Velliv_m | 71.62 | Velliv_m |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 93.18 | mix_h_a | 71.62 | PFA_h | 54.84 | PFA_h | 39.30 | PFA_h | 2.28 | PFA_h |
| 87.18 | mix_h_b | 71.24 | mix_h_a | 43.90 | mix_h_a | 22.71 | mix_h_a | 0.59 | Velliv_h |
| 85.28 | PFA_h | 65.20 | mix_h_b | 42.17 | mix_h_b | 22.12 | mix_h_b | 0.25 | mix_h_a |
| 64.53 | Velliv_h | 44.83 | Velliv_h | 29.08 | Velliv_h | 17.33 | Velliv_h | 0.11 | mix_h_b |
| 59.42 | mix_m_a | 22.36 | PFA_m | 5.20 | Velliv_m | 1.45 | Velliv_m | 0.03 | Velliv_m_l |
| 58.58 | PFA_m | 22.28 | mix_m_a | 4.56 | Velliv_m_l | 1.16 | Velliv_m_l | 0.01 | PFA_m |
| 48.34 | mix_m_b | 18.01 | Velliv_m_l | 4.45 | PFA_m | 0.59 | PFA_m | 0.00 | Velliv_m |
| 44.61 | Velliv_m_l | 16.10 | Velliv_m | 3.77 | mix_m_a | 0.22 | mix_m_a | 0.00 | mix_m_a |
| 39.14 | Velliv_m | 11.58 | mix_m_b | 1.27 | mix_m_b | 0.07 | mix_m_b | 0.00 | mix_m_b |
Summary for fit of log returns to an F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape
parameter.
xi is the estimated skewness parameter.
| Velliv_medium | Velliv_medium_long | Velliv_high | PFA_medium | PFA_high | mix_medium | mix_high | |
|---|---|---|---|---|---|---|---|
| m | 0.048 | 0.052 | 0.065 | 0.058 | 0.084 | 0.059 | 0.082 |
| s | 0.120 | 0.115 | 0.150 | 0.123 | 0.121 | 0.088 | 0.071 |
| nu | 3.304 | 2.706 | 3.144 | 2.265 | 3.185 | 2.773 | 89.863 |
| xi | 0.034 | 0.505 | 0.002 | 0.477 | 0.018 | 0.029 | 0.770 |
| R-squared | 0.993 | 0.978 | 0.991 | 0.991 | 0.964 | 0.890 | 0.961 |
| m | ranking | s | ranking | R-squared | ranking |
|---|---|---|---|---|---|
| 0.084 | PFA_high | 0.071 | mix_high | 0.993 | Velliv_medium |
| 0.082 | mix_high | 0.088 | mix_medium | 0.991 | Velliv_high |
| 0.065 | Velliv_high | 0.115 | Velliv_medium_long | 0.991 | PFA_medium |
| 0.059 | mix_medium | 0.120 | Velliv_medium | 0.978 | Velliv_medium_long |
| 0.058 | PFA_medium | 0.121 | PFA_high | 0.964 | PFA_high |
| 0.052 | Velliv_medium_long | 0.123 | PFA_medium | 0.961 | mix_high |
| 0.048 | Velliv_medium | 0.150 | Velliv_high | 0.890 | mix_medium |
Monte Carlo simulations of portfolio index values (currency
values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that
reach 0 at some point. All subsequent values for a path are set to 0, if
the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is
calculated as the share of paths finishing below index 100.
## Number of paths: 10000
| Velliv_m | Velliv_m_l | Velliv_h | PFA_m | PFA_h | mix_m_a | mix_m_b | mix_h_a | mix_h_b | |
|---|---|---|---|---|---|---|---|---|---|
| mc_m | 278.995 | 293.384 | 404.855 | 321.882 | 555.222 | 324.055 | 300.815 | 498.653 | 479.218 |
| mc_s | 123.583 | 118.147 | 217.274 | 106.531 | 244.751 | 98.254 | 82.422 | 155.916 | 164.197 |
| mc_min | 9.812 | 0.785 | 0.010 | 0.010 | 0.765 | 3.973 | 42.661 | 150.860 | 55.397 |
| mc_max | 929.858 | 2029.133 | 1713.743 | 1121.574 | 1828.260 | 723.752 | 1008.214 | 1463.716 | 1335.395 |
| dao_pct | 0.000 | 0.000 | 0.010 | 0.010 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| losing_pct | 4.720 | 3.110 | 4.300 | 1.940 | 0.950 | 1.130 | 0.340 | 0.000 | 0.120 |
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | losing_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 555.222 | PFA_h | 82.422 | mix_m_b | 150.860 | mix_h_a | 2029.133 | Velliv_m_l | 0.00 | Velliv_m | 0.00 | mix_h_a |
| 498.653 | mix_h_a | 98.254 | mix_m_a | 55.397 | mix_h_b | 1828.260 | PFA_h | 0.00 | Velliv_m_l | 0.12 | mix_h_b |
| 479.218 | mix_h_b | 106.531 | PFA_m | 42.661 | mix_m_b | 1713.743 | Velliv_h | 0.00 | PFA_h | 0.34 | mix_m_b |
| 404.855 | Velliv_h | 118.147 | Velliv_m_l | 9.812 | Velliv_m | 1463.716 | mix_h_a | 0.00 | mix_m_a | 0.95 | PFA_h |
| 324.055 | mix_m_a | 123.583 | Velliv_m | 3.973 | mix_m_a | 1335.395 | mix_h_b | 0.00 | mix_m_b | 1.13 | mix_m_a |
| 321.882 | PFA_m | 155.916 | mix_h_a | 0.785 | Velliv_m_l | 1121.574 | PFA_m | 0.00 | mix_h_a | 1.94 | PFA_m |
| 300.815 | mix_m_b | 164.197 | mix_h_b | 0.765 | PFA_h | 1008.214 | mix_m_b | 0.00 | mix_h_b | 3.11 | Velliv_m_l |
| 293.384 | Velliv_m_l | 217.274 | Velliv_h | 0.010 | Velliv_h | 929.858 | Velliv_m | 0.01 | Velliv_h | 4.30 | Velliv_h |
| 278.995 | Velliv_m | 244.751 | PFA_h | 0.010 | PFA_m | 723.752 | mix_m_a | 0.01 | PFA_m | 4.72 | Velliv_m |
| vmr | vhr | pmr | phr | mmr | mhr | |
|---|---|---|---|---|---|---|
| m | 0.064 | 0.077 | 0.061 | 0.085 | 0.062 | 0.081 |
| s | 0.081 | 0.099 | 0.063 | 0.101 | 0.048 | 0.070 |
Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | |
|---|---|---|---|---|---|---|
| P_norm(X_min) | 0.571 | 0.758 | 0.511 | 1.676 | 5.971 | 6.842 |
| P_norm(X_max) | 13.230 | 11.876 | 12.922 | 15.359 | 9.628 | 6.429 |
| P_t(X_min) | 5.377 | 5.457 | 3.489 | 4.315 | 10.570 | 8.015 |
| P_t(X_max) | 0.118 | 0.001 | 2.825 | 0.188 | 0.488 | 5.141 |
Average number of years between min or max events (respectively):
| vmr | vhr | pmr | phr | mmr | mhr | |
|---|---|---|---|---|---|---|
| norm: avg yrs btw min | 175.248 | 131.911 | 195.568 | 59.669 | 16.748 | 14.616 |
| norm: avg yrs btw max | 7.559 | 8.420 | 7.739 | 6.511 | 10.386 | 15.556 |
| t: avg yrs btw min | 18.596 | 18.324 | 28.663 | 23.173 | 9.461 | 12.476 |
| t: avg yrs btw max | 848.548 | 178349.076 | 35.400 | 531.552 | 205.104 | 19.450 |
\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?
\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]
\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).
Definition: R = 1+r
## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.
Then,
## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.
Average of returns:
## 0.5 * (R_x + R_y) = 1
So here the value of the pf at t=1 should be unchanged from t=0:
## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300
But this is clearly not the case:
## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175
Therefore we should take returns of average, not average of returns!
Let’s take the average of log returns instead:
## 0.5 * (log(R_x) + log(R_y)) = -0.143841
We now get:
## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076
So taking the average of log returns doesn’t work either.
Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.
## m(data_x): -0.008375401
## s(data_x): 0.4184349
## m(data_y): 9.445322
## s(data_y): 2.665942
##
## m(data_x + data_y): 4.718473
## s(data_x + data_y): 1.429784
m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.
| m_a | m_b | s_a | s_b |
|---|---|---|---|
| 94.308 | 94.032 | 5.930 | 6.447 |
| 94.046 | 94.419 | 6.020 | 6.583 |
| 94.537 | 94.423 | 6.064 | 6.567 |
| 94.585 | 94.491 | 6.202 | 6.386 |
| 94.470 | 94.184 | 6.192 | 6.242 |
| 94.480 | 94.510 | 6.141 | 6.366 |
| 94.484 | 94.503 | 5.843 | 6.303 |
| 94.501 | 94.179 | 5.840 | 6.397 |
| 94.314 | 94.193 | 5.941 | 6.350 |
| 94.222 | 94.482 | 6.180 | 6.661 |
## m_a m_b s_a s_b
## Min. :94.05 Min. :94.03 Min. :5.840 Min. :6.242
## 1st Qu.:94.31 1st Qu.:94.19 1st Qu.:5.933 1st Qu.:6.354
## Median :94.48 Median :94.42 Median :6.042 Median :6.391
## Mean :94.39 Mean :94.34 Mean :6.035 Mean :6.430
## 3rd Qu.:94.50 3rd Qu.:94.49 3rd Qu.:6.170 3rd Qu.:6.537
## Max. :94.59 Max. :94.51 Max. :6.202 Max. :6.661
_a and _b are very close to equal.
We attribute the differences to differences in estimating the
distributions in version a and b.
The final state is independent of the order of the preceding steps:
So does the order of the steps in the two processes matter, when mixing simulated returns?
The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.
Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]
var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618
Our distribution estimate is based on 13 observations. Is that enough
for a robust estimate? What if we suddenly hit a year like 2008? How
would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.
## m s
## Min. :0.05940 Min. :0.04936
## 1st Qu.:0.06631 1st Qu.:0.05986
## Median :0.07033 Median :0.06726
## Mean :0.07071 Mean :0.06858
## 3rd Qu.:0.07282 3rd Qu.:0.07670
## Max. :0.08455 Max. :0.09120
xiThe fit for mhr has the highest xi value of
all. This suggests right-skew:
If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).
If not, \(X\) doesn’t have a \(p\)’th moment.
See Taleb: The Statistical Consequences Of Fat Tails, p. 192
Comments
(Ignoring
mhr_a…)mhrhas some nice properties:- It has a relatively high
nuvalue of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds havenuvalues close to 3, exceptphrwhich is even worse at close to 2. (Note that for a Gaussian,nuis infinite.)- It has the lowest losing percentage of all simulations, which is better than 1/6 that of
phr.- It has a DAO percentage of 0, which is the same as
mmr, and less thanphr.- Only
phrhas a highermc_m.- It has a smaller
mc_sthan the individual components,vhrandphr.- It has the highest
xiof all fits, suggesting less left skewness. Density plots forvmr,phrandmmrhave an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely lowxivalues. The density plot formhris by far the most symmetrical of all the fits. As seen in the section “Compare Gaussian and skewed t-distribution fits”, the other skewed t-distribution fits don’t capture the max observed returns at all.- Only
mmrhas as highermc_min. However, that ofmmris 18 times higher with 62, sommris a clear winner here.- Naturally, it has a
mc_maxsmaller than the individual components,vhrandphr, but ca. 1.5 times higher thenmmr.- All the first 4 moments converge nicely. For all other fits, the 4th moment doesn’t seem to converge.
Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.
And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”